Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties

A classical (or quantum) superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n-1 functionally independent constants of the motion polynomial in the momenta, the maximum possible. If the constants are all quadratic the system is second-order superintegrable. The Kepler-Coulomb system is the best known example. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions and with QES systems. For complex Riemannian spaces with n = 2 the structure and classification of second-order superintegrable systems is complete. Here, however, we present a new and conceptually simpler approach to the classification for complex Euclidean 2-space in which the possible superintegrable systems with nondegenerate potentials correspond to points on an algebraic variety. Specifically, we determine a variety in six variables subject to two cubic and one quartic polynomial constraints. Each point on the variety corresponds to a superintegrable system. The Euclidean group E(2,ℂ) acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation.